Abstract
Two classical theorems in matrix theory, due to Schur and Horn, relate the eigenvalues of a self-adjoint matrix to the diagonal entries. These have recently been given a formulation in the setting of operator algebras as the Schur–Horn problem, where matrix algebras and diagonals are replaced respectively, by finite factors and maximal Abelian self-adjoint subalgebras (masas). There is a special case of the problem, called the carpenter problem, which can be stated as follows: for a masa $A$ in a finite factor $M$ with conditional expectation $\mathbb{E}_{A}$, can each $x\in A$ with $0\leq x\leq1$ be expressed as $\mathbb{E}_{A}(p)$ for a projection $p\in M$?
In this paper, we investigate these problems for various masas. We give positive solutions for the generator and radial masas in free group factors, and we also solve affirmatively a weaker form of the Schur–Horm problem for the Cartan masa in the hyperfinite factor.
Citation
Kenneth J. Dykema. Junsheng Fang. Donald W. Hadwin. Roger R. Smith. "The carpenter and Schur–Horn problems for masas in finite factors." Illinois J. Math. 56 (4) 1313 - 1329, Winter 2012. https://doi.org/10.1215/ijm/1399395834
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